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Weibull Models

D. N. PRABHAKAR MURTHY

University of Queensland

Department of Mechanical Engineering

Brisbane, Australia

MIN XIE

National University of Singapore

Department of Industrial and Systems Engineering

Kent Ridge Crescent, Singapore

RENYAN JIANG

University of Toronto

Department of Mechanical and Industrial Engineering

Toronto, Ontario, Canada

A JOHN WILEY & SONS, INC., PUBLICATION

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Weibull Models

WILEY SERIES IN PROBABILITY AND STATISTICS

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Weibull Models


University of Queensland

Department of Mechanical Engineering

Brisbane, Australia

MIN XIE

National University of Singapore

Department of Industrial and Systems Engineering

Kent Ridge Crescent, Singapore

RENYAN JIANG

University of Toronto

Department of Mechanical and Industrial Engineering

Toronto, Ontario, Canada

A JOHN WILEY & SONS, INC., PUBLICATION

Copyright # 2004 by John Wiley & Sons, Inc. All rights reserved.

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Library of Congress Cataloging-in-Publication Data:

Murthy, D. N. P.

Weibull models / D.N. Prabhakar Murthy, Min Xie, Renyan Jiang.

p. cm. – (Wiley series in probability and statistics)

Includes bibliographical references and index.

ISBN 0-471-36092-9 (cloth)

1. Weibull distribution. I. Xie, M. (Min) II. Jiang, Renyan, 1956-

III. Title. IV. Series.

QA273.6.M87 2004

519.204–dc21 2003053450

Printed in the United States of America.

10 9 8 7 6 5 4 3 2 1

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Dedicated to our Families

Contents

Preface xiii

PART A OVERVIEW 1

Chapter 1 Overview 3

1.1 Introduction, 3

1.2 Illustrative Problems, 5

1.3 Empirical Modeling Methodology, 7

1.4 Weibull Models, 9

1.5 Weibull Model Selection, 11

1.6 Applications of Weibull Models, 12

1.7 Outline of the Book, 15

1.8 Notes, 16

Exercises, 16

Chapter 2 Taxonomy for Weibull Models 18

2.1 Introduction, 18

2.2 Taxonomy for Weibull Models, 18

2.3 Type I Models: Transformation of Weibull Variable, 21

2.4 Type II Models: Modification/Generalization of Weibull Distribution, 23

2.5 Type III Models: Models Involving Two or More Distributions, 28

2.6 Type IV Models: Weibull Models with Varying Parameters, 30

2.7 Type V Models: Discrete Weibull Models, 33

2.8 Type VI Models: Multivariate Weibull Models, 34

2.9 Type VII Models: Stochastic Point Process Models, 37

Exercises, 39

vii

PART B BASIC WEIBULL MODEL 43

Chapter 3 Model Analysis 45


3.2 Basic Concepts, 45

3.3 Standard Weibull Model, 50

3.4 Three-Parameter Weibull Model, 54

3.5 Notes, 55

Exercises, 56

Chapter 4 Parameter Estimation 58


4.2 Data Types, 58

4.3 Estimation: An Overview, 60

4.4 Estimation Methods and Estimators, 61

4.5 Two-Parameter Weibull Model: Graphical Methods, 65

4.6 Standard Weibull Model: Statistical Methods, 67


Exercises, 82

Chapter 5 Model Selection and Validation 85


5.2 Graphical Methods, 86

5.3 Goodness-of-Fit Tests, 89

5.4 Model Discrimination, 93

5.5 Model Validation, 94

5.6 Two-Parameter Weibull Model, 95


Exercises, 100

PART C TYPES I AND II MODELS 103

Chapter 6 Type I Weibull Models 105


6.2 Model I(a)-3: Reflected Weibull Distribution, 106

6.3 Model I(a)-4: Double Weibull Distribution, 108

6.4 Model I(b)-1: Power Law Transformation, 109

viii CONTENTS

6.5 Model I(b)-2: Log Weibull Transformation, 111

6.6 Model I(b)-3: Inverse Weibull Distribution, 114

Exercises, 119

Chapter 7 Type II Weibull Models 121


7.2 Model II(a)-1: Pseudo-Weibull Distribution, 122

7.3 Model II(a)-2: Stacy–Mihram Model, 124

7.4 Model II(b)-1: Extended Weibull Distribution, 125

7.5 Model II(b)-2: Exponentiated Weibull Distribution, 127

7.6 Model II(b)-3: Modified Weibull Distribution, 134

7.7 Models II(b)4–6: Generalized Weibull Family, 138

7.8 Model II(b)-7: Three-Parameter Generalized Gamma, 140

7.9 Model II(b)-8: Extended Generalized Gamma, 143

7.10 Models II(b)9–10: Four- and Five-Parameter Weibulls, 145

7.11 Model II(b)-11: Truncated Weibull Distribution, 146

7.12 Model II(b)-12: Slymen–Lachenbruch Distributions, 148

7.13 Model II(b)-13: Weibull Extension, 151

Exercises, 154

PART D TYPE III MODELS 157

Chapter 8 Type III(a) Weibull Models 159


8.2 Model III(a)-1: Weibull Mixture Model, 160

8.3 Model III(a)-2: Inverse Weibull Mixture Model, 176

8.4 Model III(a)-3: Hybrid Weibull Mixture Models, 179

8.5 Notes, 179

Exercises, 180

Chapter 9 Type III(b) Weibull Models 182


9.2 Model III(b)-1: Weibull Competing Risk Model, 183

9.3 Model III(b)-2: Inverse Weibull Competing Risk Model, 190

9.4 Model III(b)-3: Hybrid Weibull Competing Risk Model, 191

9.5 Model III(b)-4: Generalized Competing Risk Model, 192

Exercises, 195

CONTENTS ix

Chapter 10 Type III(c) Weibull Models 197


10.2 Model III(c)-1: Multiplicative Weibull Model, 198

10.3 Model III(c)-2: Inverse Weibull Multiplicative Model, 203

Exercises, 206

Chapter 11 Type III(d) Weibull Models 208


11.2 Analysis of Weibull Sectional Models, 210

11.3 Parameter Estimation, 216

11.4 Modeling Data Set, 219

11.5 Applications, 219

Exercises, 220

PART E TYPES IV TO VII MODELS 221

Chapter 12 Type IV Weibull Models 223


12.2 Type IV(a) Models, 224

12.3 Type IV(b) Models: Accelerated Failure Time (AFT) Models, 225

12.4 Type IV(c) Models: Proportional Hazard (PH) Models, 229

12.5 Model IV(d)-1, 231

12.6 Type IV(e) Models: Random Parameters, 232

12.7 Bayesian Approach to Parameter Estimation, 236

Exercises, 236

Chapter 13 Type V Weibull Models 238


13.2 Concepts and Notation, 238

13.3 Model V-1, 239

13.4 Model V-2, 242

13.5 Model V-3, 243

13.6 Model V-4, 244

Exercises, 245

Chapter 14 Type VI Weibull Models (Multivariate Models) 247


x CONTENTS

14.2 Some Preliminaries and Model Classification, 248

14.3 Bivariate Models, 250

14.4 Multivariate Models, 256

14.5 Other Models, 258

Exercises, 258

Chapter 15 Type VII Weibull Models 261


15.2 Model Formulations, 261

15.3 Model VII(a)-1: Power Law Process, 265

15.4 Model VII(a)-2: Modulated Power Law Process, 272

15.5 Model VII(a)-3: Proportional Intensity Model, 273

15.6 Model VII(b)-1: Ordinary Weibull Renewal Process, 274

15.7 Model VII(b)-2: Delayed Renewal Process, 277

15.8 Model VII(b)-3: Alternating Renewal Process, 278

15.9 Model VII(c): Power Law–Weibull Renewal Process, 278

Exercises, 278

PART F WEIBULL MODELING OF DATA 281

Chapter 16 Weibull Modeling of Data 283


16.2 Data-Related Issues, 284

16.3 Preliminary Model Selection and Parameter Estimation, 285

16.4 Final Model Selection, Parameter Estimation, and Model Validation, 287

16.5 Case Studies, 290

16.6 Conclusions, 299

Exercises, 299

PART G APPLICATIONS IN RELIABILITY 301

Chapter 17 Modeling Product Failures 303


17.2 Some Basic Concepts, 304

17.3 Product Structure, 306

17.4 Modeling Failures, 306

17.5 Component-Level Modeling (Black-Box Approach), 306

CONTENTS xi

17.6 Component-Level Modeling (White-Box Approach), 308

17.7 Component-Level Modeling (Gray-Box Approach), 312

17.8 System-Level Modeling (Black-Box Approach), 313

17.9 System-Level Modeling (White-Box Approach), 316

Chapter 18 Product Reliability and Weibull Models 324


18.2 Premanufacturing Phase, 325

18.3 Manufacturing Phase, 332

18.4 Postsale Phase, 336

18.5 Decision Models Involving Weibull Failure Models, 341

References 348Index 377

xii CONTENTS

Preface

Mathematical models have been used in solving real-world problems from many

different disciplines. This requires building a suitable mathematical model. Two

different approaches to building mathematical models are as follows:

1. Theory-Based Modeling Here, the modeling is based on theories (from

physical, biological, and social sciences) relevant to the problem. This kind of

model is also called a physics-based model or white-box model.

2. Empirical Modeling Here the data available forms the basis for model

building, and it does not require an understanding of the underlying

mechanisms involved. This kind of model is also called as data-dependent

model or black-box model.

In the black-box approach to modeling, one first carries out an analysis of the

data, and then one determines the type of mathematical formulation appropriate to

model the data.

Many data exhibit a high degree of variability or randomness. These kinds of

data are often best modeled by a suitable probability model (such as a distribution

function) so that the data can be viewed as observed outcomes (values) of random

variables from the distribution.

Black-box modeling is a multistep process. It requires a good understanding of

probability and of statistical inference. In probability, we use the model to make

statements about the nature of the data that may result if the model is correct. This

involves model analysis using analytical and simulation techniques. The principal

objective of statistical inference is to use the available data to make statements

about the probability model, either in terms of probability distribution itself or in

terms of its parameters or some other characteristics. This involves topics such as

model selection, estimation of model parameters, and model validation. As a result,

probability and statistical inference may be thought of as inverses of one another as

indicated below.

xiii

Procedures of statistical inference are the basic tools of data analysis. Most are

based on quite specific assumptions regarding the nature of the probabilistic

mechanism that gave rise to the data.

Many standard probability distribution functions (e.g., normal, exponential) have

been used as models to model data exhibiting significant variability. More complex

models are distributions derived from standard distributions (e.g., lognormal). One

distribution of particular significance is the Weibull distribution. It is named after

Professor Waloddi Weibull who was the first to promote the usefulness of this to

model data sets of widely differing characteristics.

Over the last two decades several new models have been proposed that are either

derived from, or in some way related to, the Weibull distributions. We use the term

Weibull models to denote such models. They provide a richness that makes them

appropriate to model complex data sets.

The literature on Weibull models is vast, disjointed, and scattered across many

different journals. There are a couple of books devoted solely to the Weibull

distribution, but these are oriented toward training and/or consulting purposes.

There is no book that deals with the different Weibull models in an integrated

manner. This book fills that gap.

The aims of this book are to:

1. Integrate the disjointed literature on Weibull models by developing a proper

taxonomy for the classification of such models.

2. Review the literature dealing with the analysis and statistical inference

(parameter estimation, goodness of fit) for the different Weibull models.

3. Discuss the usefulness of the Weibull probability paper (WPP) plot in the

model selection to model a given data set.

4. Highlight the use of Weibull models in reliability theory.

The book would be of great interest to practitioners in reliability and other

disciplines in the context of modeling data sets using univariate Weibull models.

Some of the exercises at the end of each chapter define potential topics for future

research. As such, the book would also be of great interest to researchers interested

in Weibull models.

The book is organized into the following seven parts (Parts A to G).

Part A consists of two chapters. Chapter 1 gives an overview of the book.

Chapter 2 deals with the taxonomy for Weibull models and gives the mathematical

xiv PREFACE

structure of the different models. The taxonomy involves seven different categories

that we denote as Types I to VII.

Part B consists of three chapters. Chapter 3 deals with model analysis and

discusses various model-related properties. Chapter 4 deals with parameter estima-

tion and examines different data structures, estimation methods, and their proper-

ties. Chapter 5 deals with model selection and validation, where the focus is on

deciding whether a specific model is appropriate to model a given data set or not. In

these chapters many concepts and techniques are introduced, and these are used in

later chapters. In these three chapters, the analysis, estimation, and validation are

discussed for the standard Weibull model as well as the three-parameter Weibull

model, as the two models are very similar.

Part C consists of two chapters. Chapter 6 deals with Type I models derived from

nonlinear transformations of random variables from the standard Weibull model.

Chapter 7 deals with Type II models, which are obtained by modifications of the

standard Weibull model and in some cases involving one or more additional

parameters.

Part D consists of four chapters and deals with Type III models. Chapter 8 deals

with the mixture models, Chapter 9 with the competing risk models, Chapter 10

with the multiplicative models, and Chapter 11 with the sectional models.

Part E consists of four chapters. Chapter 12 deals with Type IV models,

Chapter 13 with Type V models, Chapter 14 with Type VI models, and

Chapter 15 with Type VII models.

In Parts C to E, for each model we review the available results (analysis,

statistical inference, etc.) relating to the model.

Part F consists of a single chapter (Chapter 16) dealing with model selection to

model a given data set.

Part G deals with the application of Weibull models in reliability theory and

consists of two chapters. Chapter 17 deals with modeling failures. Chapter 18

discusses a variety of reliability-related decision problems in the different phases

(premanufacturing, manufacturing, and postsale) of the product life cycle and

reviews the literature relating to Weibull failure models.

Reliability engineers and applied statisticians involved with reliability and

survival analysis should find this as a valuable reference book. It can be used as

a textbook for a course on probabilistic modeling at the graduate (or advanced

undergraduate) level in industrial engineering, operations research, and statistics.

We would like to thank Professor Wallace Blischke (University of Southern

California) for his comments on several chapters of the book and Dr Michael

Bulmer and Professor John Eccleston (University of Queensland) for their

contributions to Chapter 16. Several reviewers of the book have given detailed

and encouraging comments and we are grateful for their contributions. Special

thanks to Steve Quigley, Heather Bergman, Susanne Steitz, and Christine Punzo at

Wiley for their patience and support.

We would like to thank several funding agencies for their support over the last

few years. In particular, the funding from the National University of Singapore for

PREFACE xv

the first author’s appointment as a distinguished visiting professor in 1998 needs a

special mention as it lead to the initiation of this project.

Finally, the encouragement and support of our families are also greatly

appreciated.


Brisbane, Queensland, Australia

MIN XIE

Kent Ridge, Singapore

RENYAN JIANG

Toronto, Canada

xvi PREFACE

PA R T A

Overview

C H A P T E R 1

Overview

1.1 INTRODUCTION

In the real world, problems arise in many different contexts. Problem solving is an

activity that has a history as old as the human race. Models have played an impor-

tant role in problem solving and can be traced back to well beyond the recorded

history of the human race. Many different kinds of models have been used. These

include physical (full or scaled) models, pictorial models, analog models, descrip-

tive models, symbolic models, and mathematical models. The use of mathematical

models is relatively recent (roughly the last 500 years). Initially, mathematical mod-

els were used for solving problems from the physical sciences (e.g., predicting

motion of planets, timing of high and low tides), but, over the last few hundred

years, mathematical models have been used extensively in solving problems from

biological and social sciences. There is hardly any discipline where mathematical

models have not been used for solving problems.

Two different approaches to building mathematical models are as follows:

1. Theory-Based Modeling. Here, the modeling is based on the establishedtheories (from physical, biological, and social sciences) relevant to the

problem. This kind of model is also called physics-based model or white-

box model as the underlying mechanisms form the starting point for the

model building.

2. Empirical Modeling. Here, the data available forms the basis for the model

building, and it does not require an understanding of the underlying

mechanisms involved. As such, these models are used when there is

insufficient understanding to use the earlier approach. This kind of model

is also called data-dependent model or black-box model.

In empirical modeling, the type of mathematical formulations needed for mod-

eling is dictated by a preliminary analysis of data available. If the analysis indicates

Weibull Models, by D.N.P. Murthy, Min Xie, and Renyan Jiang.

ISBN 0-471-36092-9 # 2004 John Wiley & Sons, Inc.

3

that there is a high degree of variability, then one needs to use models that can

capture this variability. This requires probabilistic and stochastic models to model

a given data set.

Effective empirical modeling requires good understanding of (i) the methodology

needed for model building, (ii) properties of different models, and (iii) tools and tech-

niques to determine if a particular model is appropriate to model a given data set.

A variety of such models have been developed and studied extensively. One such

class of models is the Weibull models. These are a collection of probabilistic and

stochastic models derived from the Weibull distribution. These can be divided into

univariate and multivariate models and each, in turn, can be further subdivided into

continuous and discrete. Weibull models have been used in many different applica-

tions to model complex data sets.

1.1.1 Aims of the Book

This book deals with Weibull models and their applications in reliability. The aims

of the book are as follows:

1. Develop a taxonomy to integrate the different Weibull models.

2. Review the literature for each model to summarize model properties andother issues.

3. Discuss the use of Weibull probability paper (WPP) plots in model selection.

It allows the model builder to determine whether one or more of the Weibull

models are suitable for modeling a given data set.

4. Highlight issues that need further study.

5. Illustrate the application of Weibull models in reliability theory.

The book provides a good foundation for empirical model building involving

Weibull models. As such, it should be of interest to practitioners from many differ-

ent disciplines. The book should also be of interest to researchers as some topics for

future research are defined as part of the exercises at the end of several chapters.

1.1.2 Outline of Chapter

The outline of the chapter is as follows. We start with a collection of real-world

problems in Section 1.2 and discuss the data aspects and empirical models to obtain

solutions to the problems. Section 1.3 deals with the modeling methodology, and

we discuss the different issues involved. We highlight the role of statistics, probabil-

ity theory, and stochastic processes in the context of the link between data and model.

Section 1.4 starts a brief historical perspective and then introduces the standard

Weibull model (involving the two-parameter Weibull distribution). Following

this, a taxonomy to classify the different Weibull models is briefly discussed. Given

a univariate continuous data set, a question of great interest to model builders is

whether one of the Weibull models is suitable for modeling the given data set or

not. This topic is discussed in Section 1.5. Section 1.6 deals with the applications

of Weibull models where we start with a short list of applications to highlight the

4 OVERVIEW

diverse range of applications of the Weibull models in different disciplines. How-

ever, in this book we focus on the application of Weibull models in the context of

product reliability from a product life perspective. We discuss this briefly so as to

set the scene for the discussion on Weibull model applications later in the book. We

finally conclude with an outline of the book in Section 1.7.

1.2 ILLUSTRATIVE PROBLEMS

In this section we give a few illustrative problems and the types of data available to

build models to obtain solutions to the problems.

Example 1: Tidal HeightsAt a popular tourist beach the cyclone season precedes the tourist season. Very high

tides during the cyclone season cause the erosion of sand on the beach. The erosion

is related to the amplitude of the high tide, and it takes a long time for the beach to

recover naturally from the effect of such erosion. Often, sand needs to be pumped to

restore the loss and to ensure high tourist numbers. A problem of interest to the city

council responsible for the beach is the probability that a high tide during the

cyclone season exceeds some specified height resulting in the council incurring

the sand pumping cost. The data available is the amplitude of high tides over

several years.

Example 2: Efficacy of TreatmentIn medical science, a problem of interest is in determining the efficacy of a new

treatment to control the spread of a disease (e.g., cancer). In this case, clinical trials

are carried out for a certain period. The data available are the number entering the

program, the time instants, and the age at death for the patients who died during the

trial period, ages of the patients who survived the test period, and so on. Similar

data for a sample not given the new treatment might also be available. The problem

is to determine if the new treatment increases the life expectancy of the patients.

Example 3: Strength of ComponentsDue to manufacturing variability, the strength of a component varies significantly.

The component is used in an environment where it fails immediately when put into

use if its strength is below some specified value. The problem is to determine the

probability that a component manufactured will fail under a given environment. If

this probability is high, changing the material, the process of manufacturing, or

redesigning might be the alternatives that the manufacturer might need to explore.

The data available is the laboratory test data. Here items are subjected to increasing

levels of stress and the stress level at failure being recorded.

Example 4: Insurance ClaimsWhenever there is a legitimate claim, a car insurance company has to pay out. The

pay out indicates a high degree of variability (since it can vary from a small to a

ILLUSTRATIVE PROBLEMS 5

very large amount). The insurance company has used the expected value as the

basis for determining the annual premium it should charge its customers. It is plan-

ning to change the premium and is interested in assessing the probability of an indi-

vidual claim exceeding five times the premium charged. The data available is the

insurance claims over the last few years.

Example 5: Growth of TreesPaper manufacturing requires wood chips. One way of producing wood chips is

through plantations where trees are harvested when the trees reach a certain age.

The height of the tree at the time of the harvesting is critical as the volume of

wood chips obtained is related to this height. The heights of trees vary significantly.

As a result, the output of a plantation can vary significantly, and this has an impact

on the profitability of the operation. The operator of a plantation is faced with the

problem of choosing between two different types of trees. The data available (from

other plantations) are the heights of trees at the time of harvesting for both species.

Example 6: Maintenance of Street LightsThe life of electric bulbs used for street lighting is uncertain and is influenced by a

variety of factors (variability in the material used and in the manufacturing process,

fluctuations in the voltage, etc.). Replacement of an individual failed item is in gen-

eral expensive. In this case the road authority might decide on some preventive

maintenance action where the bulbs are replaced by new ones at set time instants

t ¼ kT ; k ¼ 1; 2; . . . : The cost of replacing a bulb under such a replacement policyis much cheaper, but it involves discarding the remaining useful life of the bulb.

Any failure in between results in the failed item being replaced by a new one at

a much higher cost. The problem facing the authority is to determine the optimal

T that minimizes the expected cost. The data available is the historical record of

failures and preventive replacements in the past.

Example 7: Stress on Offshore PlatformAn offshore platform must be designed to withstand the buffeting of waves. The

impact of each wave on the structure is determined by the energy contained in

the wave. The wave height is an indicator of the energy in a wave. The data avail-

able are the heights of successive waves over a certain time interval, and this exhi-

bits a high degree of variability. The problem is to determine the risk of an offshore

platform collapsing if designed to withstand waves up to a certain height.

Example 8: Wind VelocityWindmills are structures that harness the energy in the wind and convert it into elec-

trical or mechanical energy. The wind velocity fluctuates, and as a result the output

of the windmill fluctuates. The economic viability of a windmill is dependent on

it being capable of generating a certain minimum level of output for a specified

fraction of the day. The problem is to determine the viability of windmills based

on the data for wind speeds measured every 5 minutes over a week.

6 OVERVIEW

Example 9: Rock BlastingMining involves blasting ore formation using explosives. The effect of explosion is

that it fragments the ore into different sizes. Ore smaller than the minimum accep-

table size is of no value as it are unsuited for processing. Ore lumps bigger than the

maximum acceptable size need to be broken down, which involves additional cost.

The problem of interest to a mine operator is to determine the size distribution

of ore under different blasting strategies so as to decide on the best blasting strategy.

In this case, the data available are the size distribution of ore randomly sampled

after a blast.

Example 10: Spare Part PlanningFor commercial equipment (e.g., aircraft, locomotive) downtime implies a loss of

revenue. Downtime occurs due to failure of one or more components of the equip-

ment. Failure of a component is dependent on the reliability of the component. The

downtime is dependent on whether a spare is available or not and the time to get a

spare if one is not available. When the component is expensive, one must manage

the inventory of spare parts properly. Carrying a large inventory implies too much

capital being tied up. On the other hand, having a small inventory can lead to high

downtimes. The problem is to determine the optimal spare part inventory for com-

ponents. The data available are the failure times for the different components over a

certain period of time.

1.3 EMPIRICAL MODELING METHODOLOGY

The empirical modeling process involves the following five steps:

Step 1: Collecting data

Step 2: Analysis of data

Step 3: Model selection

Step 4: Parameter estimation

Step 5: Model validation

In this section we briefly discuss each of these steps.

Step 1: Collecting DataData can be either laboratory data or field data. Laboratory data is often obtained

under controlled environment and based on a properly planned experiment. In con-

trast, field data suffers from variability in the operation environment as well as other

uncontrollable factors.

The form of data can vary. In the case of reliability data, it could be continuous

valued (e.g., life of an individual item) or discrete valued (e.g., number of items

failing in a specified interval). In the former case, it could represent failure

times or censored times (the lives of nonfailed items when data collection was

stopped) for items. We shall discuss this issue in greater detail in Chapter 4.

EMPIRICAL MODELING METHODOLOGY 7

Finally, when the data needed for modeling is not available, one needs to collect

data based on a proper experiment on expert judgment in some cases. The experi-

ment, in general, is discipline specific. We will discuss this issue in the context of

product reliability later in the book.

Step 2: Preliminary Analysis of DataGiven a data set, one starts with a preliminary analysis of the data. Suppose that the

data set available is given by ðt1; t2; . . . ; tnÞ. In the first stage, one computes varioussample statistics (such as max, min, mean, sample variance, median, and first and

third quartiles) based on the data. If the range (¼ max – min) is small relative to thesample mean, one might ignore the variability in the data and model the data by the

sample mean. However, when this is not the case, then the model needs to mimic

this variability in the data. In the case of time-ordered data, preliminary analysis is

used to determine properties such as trends (increasing or decreasing), correlation

over time, and so forth.

The main aim of the analysis is to assist in determining whether a particular

model is appropriate or not to model a given data set. Many different plots have

been developed to assist in this. Some of these plots (e.g., histogram) are general

and others (e.g., Weibull probability paper plot) were originally developed for a

particular model but have since been used for a broader class of models.

Step 3: Model SelectionSuppose that the data set ðt1; t2; . . . ; tnÞ exhibits significant variability. In this casethe data set needs to be viewed as an observed value of a set of random variables

ðT1; T2; . . . ; TnÞ. If the random variables are statistically independent, then each Tcan be modeled by a univariate probability distribution function:

Fðt; yÞ ¼ PðT � tÞ �1 < t < 1 ð1:1Þwhere y denotes the set of parameters for the distribution. In some cases the rangeof t is constrained. For example, if T represents the lifetime of an item, then it is

constrained to be nonnegative so that Fðt; yÞ is zero for t < 0.Model selection involves choosing an appropriate model formulation (e.g., a dis-

tribution function) to model a given data set. In order to execute this step, one needs

to have a good understanding of the properties of different model formulations

suitable for modeling. Some basic concepts are discussed in Chapter 3. Probability

theory deals with such study for a variety of model formulations. An important

feature of modeling is that often there is more than one model formulation that

will adequately model a given data set. In other words, one can have multiple

models for a given data set.

The data source often provides a clue to the selection of an appropriate model. In

the case of failure data, for example, lognormal or Weibull distributions have been

used for modeling failures due to fatigue and exponential distributions for failure of

electronic components. In order to use this knowledge, the model builder must be

familiar with earlier models for failures of different items.

If the data are not independent, one needs to use models involving multivariate

distribution functions. If time is a factor that needs to be included in the model

8 OVERVIEW

explicitly, then the model becomes more complex. The building of such models

requires concepts from stochastic processes.

Step 4: Parameter EstimationOnce a model is selected, one needs to estimate the model parameters. The esti-

mates are obtained using the data available. A variety of techniques have been

developed, and these can be broadly divided into two categories—graphical and

analytical. The accuracy of the estimate is dependent on the size of the data

and the method used. Graphical methods yield crude estimates while analytical methods

yield better estimates and confidence limits for the estimates. The basic concepts are

discussed in in Chapter 4 and in later chapters in the context of specific models.

Step 5: Model ValidationOne can always fit a model to a given data set. However, the model might not be

appropriate or adequate. An inappropriate model, in general, will not yield the

desired solution to the problem. Hence, it is necessary to check the validity of

the model selected. There are several methods for doing this. The basic concepts

are discussed in Chapter 5 and in later chapters in the context of specific models.

Comments

1. Steps 2, 4, and 5 deal with statistical inference. In probability theory, one

models the uncertainty (randomness) through a distribution function, and then

makes statements, based on the model, about the nature (e.g., variability) of

the data that may result if the model is correct. The principal objective of

statistical inference is to use data to make statements about the model, either

in terms of probability distribution itself or in terms of its parameters or some

other characteristics. Thus, probability theory and statistical inference may be

thought of as inverse of one another as indicated:

Probability theory: Model ! DataStatistics: Data ! Model

2. Statistical inference requires concepts, tools, and techniques from the theory

of statistics. Understanding a model requires studying the properties of the

model. This requires concepts, tools, and techniques from the theory of

probability and the theory of stochastic processes.

3. In this book we discuss both model properties and statistical inference forWeibull models.

1.4 WEIBULL MODELS

1.4.1 Historical Perspective

The three-parameter Weibull distribution is given by the distribution function

Fðt; yÞ ¼ 1� exp � t � ta

� �b� �t � t ð1:2Þ

WEIBULL MODELS 9

The parameters of the distribution are given by the set y ¼ fa; b; tg witha > 0;b > 0, and t � 0. The parameters a; b, and t are the scale, shape, and loca-tion parameters of the distribution, respectively. The distribution is named after

Waloddi Weibull who was the first to promote the usefulness of this to model

data sets of widely differing character. The initial study by Weibull (Weibull,

1939) appeared in a Scandinavian journal and dealt with the strength of materials.

A subsequent study in English (Weibull, 1951) was a landmark work in which he

modeled data sets from many different disciplines and promoted the versatility of

the model in terms of its applications in different disciplines.

A similar model was proposed earlier by Rosen and Rammler (1933) in the con-

text of modeling the variability in the diameter of powder particles being greater

than a specific size. The earliest known publication dealing with the Weibull distri-

bution is a work by Fisher and Tippet (1928) where this distribution is obtained as

the limiting distribution of the smallest extremes in a sample. Gumbel (1958) refers

to the Weibull distribution as the third asymptotic distribution of the smallest

extremes.

Although Weibull was not the first person to propose the distribution, he was

instrumental in its promotion as a useful and versatile model with a wide range

of applicability. A report by Weibull (Weibull, 1977) lists over 1000 references

to the applications of the basic Weibull model, and a recent search of various data-

bases indicate that this has increased by a factor of 3 to 4 over the last 30 years.

1.4.2 Taxonomy

The two-parameter Weibull distribution is a special case of (1.2) with t ¼ 0 so that

Fðt; yÞ ¼ 1� exp � ta

� �b� �t � 0 ð1:3Þ

We shall refer to this as the standard Weibull model with að> 0Þ and bð> 0Þ beingthe scale and shape parameters respectively. The model can be written in alternate

parametric forms as indicated below:

Fðt; yÞ ¼ 1� exp �ðltÞbh i

ð1:4Þ

with l ¼ 1=a;

Fðt; yÞ ¼ 1� exp � tb

a0

� �ð1:5Þ

with a0 ¼ ab; and

Fðt; yÞ ¼ 1� expð�l0tbÞ ð1:6Þ

10 OVERVIEW

with l0 ¼ ð1=aÞb. Although they are all equivalent, depending on the context a par-ticular parametric representation might be more appropriate. In the remainder of the

book, the form for the standard Weibull model is (1.3) unless indicated otherwise.

A variety of models have evolved from this standard model. We propose a tax-

onomy for classifying these models, and it involves seven major categories labeled

Types I to VII. In this section, we briefly discuss the basis for the taxonomy, and the

different models in each category are discussed in Chapter 2.

Let T denote the random variable from the standard Weibull model. Let the dis-

tribution function for the derived model be Gðt; yÞ, and let Z denote the randomvariable from this distribution. The links between the standard Weibull model

and the seven different categories of Weibull models are as follows:

Type I Models Here Z and T are related by a transformation. The transformation

can be either (i) linear or (ii) nonlinear.

Type II Models Here Gðt; yÞ is related to Fðt; yÞ through some functional rela-tionship.

Type III Models These are univariate models derived from two or more distribu-

tions with one or more being a standard Weibull distribution. As a result, Gðt; yÞ is aunivariate distribution function involving one or more standard Weibull distribu-

tions.

Type IV Models The parameters of the standard Weibull model are constant. For

models belonging to this group, this is not the case. As a result, they are either

a function of the variable t or some other variables (such as stress level) or are

random variables.

Type V Models In the standard Weibull model, the variable t is continuous valued

and can assume any value in the interval ½0;1Þ. As a result, T is a continuous ran-dom variable. In contrast, for Type V models Z can only assume nonnegative inte-

ger values, and this defines the support for Gðt; yÞ.

Type VI Models The standard Weibull model is a univariate model. Type VImodels are multivariate extensions of the standard Weibull model. As a result,

Gð�Þ is a multivariate function of the form Gðt1; t2; . . . ; tnÞ and related to the stan-dard Weibull in some manner.

Type VII Models These are stochastic point process models with links to the

standard Weibull model.

1.5 WEIBULL MODEL SELECTION

Model selection tends to be a trial-and-error process. For Types I to III models the

Weibull probability paper plot provides a systematic procedure to determine

WEIBULL MODEL SELECTION 11

whether one of these models is suitable for modeling a given data set or not. It is

based on the Weibull transformations

y ¼ ln � ln½1� FðtÞ�f g and x ¼ lnðtÞ ð1:7Þ

A plot of y versus x is called the Weibull probability plot. In the early 1970s a spe-

cial paper was developed for plotting the data under this transformation and was

referred to as the Weibull probability paper (WPP) and the plot called the WPP

plot. These days, most reliability software packages contain programs to produce

these plots automatically given a data set. We use the term WPP plot to denote

the plot using computer packages.

1.6 APPLICATIONS OF WEIBULL MODELS

Weibull models have been used in many different applications and for solving a

variety of problems from many different disciplines. Table 1.1 gives a small sample

of the application of Weibull models along with references where interested readers

can find more details.

1.6.1 Reliability Applications

All man-made systems (ranging from simple products to complex systems) are

unreliable in the sense that they degrade with time and/or usage and ultimately

fail. The following material is from Blischke and Murthy (2000).

The reliability of a product (system) is the probability that the product (system)

will perform its intended function for a specified time period when operating under

normal (or stated) environmental conditions.

Product Life Cycle and ReliabilityA product life cycle (for a consumer durable or an industrial product), from the

point of view of the manufacturer, is the time from initial concept of the product

to its withdrawal from the marketplace. It involves several stages as indicated in

Figure 1.1.

The process begins with an idea to build a product to meet some customer

requirements regarding performance (including reliability) targets. This is usually

based on a study of the market and the potential demand for the product being

planned. The next step is to carry out a feasibility study. This involves evaluating

whether it is possible to achieve the targets within the specified cost limits. If this

analysis indicates that the project is feasible, an initial product design is undertaken.

A prototype is then developed and tested. It is not unusual at this stage to find that

achieved performance level of the prototype product is below the target value. In

this case, further product development is undertaken to overcome the problem.

Once this is achieved, the next step is to carry out trials to determine performance

of the product in the field and to start a preproduction run. This is required because

12 OVERVIEW

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